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Letter

Expeditious Stochastic Calculation of Random-Phase ApproximationEnergies for Thousands of Electrons in 3 Dimensions

Daniel Neuhauser, Eran Rabani, and Roi BaerJ. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/jz3021606 • Publication Date (Web): 08 Mar 2013

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1

Expeditious Stochastic Calculation of Random-Phase Approximation Ener-gies for Thousands of Electrons in 3 Dimensions

Daniel Neuhauser†, Eran Rabani‡ and Roi Baer

† Department of Chemistry and Biochemistry, University of California, Los Angeles CA 90095, USA. ‡ School of Chemistry, The Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel. Fritz Haber Center for Molecular Dynamics, Institute of Chemistry, Hebrew University, Jerusalem 91904 Israel.

ABSTRACT: A fast method is developed for calculating the Random-Phase-Approximation (RPA) correlation energy for density func-

tional theory. The correlation energy is given by a trace over a projected RPA response matrix and the trace is taken by a stochastic approach

using random perturbation vectors. For a fixed statistical error in the total energy per electron the method scales at most, quadratically with

the system size, however, in practice, due to self-averaging, requires less statistical sampling as the system grows and the performance is

close to linear scaling. We demonstrate the method by calculating the RPA correlation energy for cadmium selenide and silicon nanocrystals

with over 1500 electrons. We find that the RPA correlation energies per electron are largely independent of the nanocrystal size. In addition,

we show that a correlated sampling technique enables calculation of the energy difference between two slightly distorted configurations with

scaling and a statistical error similar to that of the total energy per electron.

Local and semi-local correlation functionals of Kohn-Sham

(KS) density functional theory (DFT) fail to describe long-

range van der Waals interactions and other types of dynam-

ical screening effects. ADDIN EN.CITE One route for over-

coming these deficiencies is RPA theory1, 3-6 based on the

KS-DFT adiabatic connection formalism7-9 in combination

with the fluctuation dissipation theorem.10 In recent years this

approach, especially when combined with exact exchange,

was used successfully for treating various ailments of KS-

DFT in molecular and condensed matter systems5, 6, 11-18.

The greatest hurdle facing widespread use of RPA is its ex-

ceedingly high computational cost. Several approaches have

been developed 5, 6, 13, 19, 20 for reducing the naïve ( ) RPA

scaling to ( ), ( is a measure of system size); however,

this is still expensive. The problem is aggravated when plane-

waves or real-space grids are used, suffering from the huge

number of unoccupied states and the strong reliance of the

RPA energy on the unoccupied energies.21-23

In the present letter, we develop a stochastic sampling meth-

od for estimating the RPA correlation energy. Related sam-

pling techniques have been recently developed by us for es-

timating the rate of multiexciton generation in nanocrystals

(NCs),24 for a linear scaling calculation of the exchange ener-

gy,25 and for overcoming the computational bottleneck in

Møller-Plesset second order perturbation theory (MP2).26

RPA, applied on top of a grid or plane waves calculation,

starts from the KS or generalized KS Hamiltonian which

can be applied to any wavefunction in linear scaling.27 For a

closed-shell system of electrons on a grid/basis of size ,

the lowest KS orbitals ( ) of are occupied and -

are unoccupied.28 The RPA correlation energy can be written

as5

∑ ( - )

, where are eigenvalues

of “ ” defined by:

( ) (

) ( ) (

) (1)

and

(2)

where ∫ ( ) (

) ( ) ( )

| - | are the Cou-

lomb integrals, is a coupling strength parameter ( is

full-strength Coulomb coupling and is the non-

interacting limit), and - is the difference of eigen-

values of . Usually, all are real; however , although

having pure imaginary eigenvalues, is a real matrix operating

on real vectors ( ). Note that are also the eigenvalues of

the matrix (

- - ) appearing in standard RPA treatments.5

An alternative formulation starts from the expression:5

[ ( ) ( ( )

)

] (3)

where ( )

∑ ( )

. However, the calculation of

( ) is still prohibitive for large systems because of the high

cost of diagonalization of the ( - ) ( - ) matrix

(in grid representations and easily reach and

respectively).

Our formulation is based on linear-response time-dependent

Hartree approach.29, 30 is still given by Eq. ‎(3) but ( )

is replaced by the following trace: 29, 30

( ) [ ( ( ))] (4)

Here ( ) ( ), where ( ) is the Heaviside step func-

tion, which we approximate as ( )

(- ); is a lin-

ear operator, revealed when linearizing the time-dependent

KS equations (see refs. 14 and 30 for details):

( ) (

( )

[ ] ( ) ) (5)

( ) and ( ) are functions, originally describing the time-

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dependent Hartree response of the -th KS orbital ( ), but

are used here as stochastic perturbations as detailed below.

[ ]( ) ∫ ( )

| - | is the Hartree perturbation poten-

tial depending linearly on the ’s via

( ) ∑ ( ) ( )

(6)

One can expand in the basis of the KS orbitals and obtain

its matrix having positive imaginary

eigenvalues and an equivalent negative set. ( - ) ei-

genvalues can be classified as “occupied-unoccupied” transi-

tions , and eigenvalues are occupied-occupied transi-

tions . Obviously, the dimensions of and differ, as the

latter describes only occupied-unoccupied transitions. None-

theless, within the occupied-unoccupied space the matrices

and eigenvalues are identical:31

(7)

( ) in Eq. ‎(4) is computed in two principal steps:

1) The trace is replaced by an average (denoted by curly

brackets) over random perturbation vectors ( ):

( ) {⟨( )| ( ( ))| (

)⟩} (8)

( ) is shorthand notation for the entire set of ’s and

’s.

2) The selection of the random vector and is done by

generating two vectors composed of random complex

phases at each grid point : ( )

( )

where is the grid spacing and ( ) is a random num-

ber between 0 and . Then one sets:

( ) ( ) ( )⟨ | ⟩ ( )⟨ | ⟩ (9)

3) The action of the operator ( ) on ( ) is performed

using an iterative modified Chebyshev polynomial expan-

sion approach, so that:

( ) ∑

(10)

where

{⟨( )| ( ( )

( ))⟩} (11)

are the modified Chebyshev residues, and ( ( )

( )) are cal-

culated iteratively:

( ( )

( ))

( ( )

( )) (

( )

( )) ( ) (12)

with

( ( )

( )) (

) (

( )

( ))

( ( )

( )) (13)

Note that is a real operator (Eq. ‎(5)) so all calculations are

done on real functions.

( - ) is half the eigenval-

ue range of . The are numerical coefficients obtained as

follows: First, prepare a series of length ,

( (

)), - and then set

( ) (for ) where { } are the discrete Fou-

rier transform of { }. The series length is chosen large

enough so that the sum in Eq. ‎(10) converges, i.e. | | is

smaller than a prescribed tolerance.

We rely on correlated sampling to reduce the statistical error

in computing . is computed for 3 values of

and – (with - ) using the same random number

seeds, and then the RPA energy is estimated by:

[ ( )

∑ (

) ( )

] (14)

Figure 1: The error of the stochastic estimate for the RPA energy per

electron with respect to the exact value (black curve) and the vari-

ance (red curve) as function of stochastic iterations for SiH4.

Table 1: Parameters for the CdSe nanocrystals NCs. Shown are the

number of Cd ( ) and Se ( ) atoms, electrons ( ), NC diame-

ter ( ), the numerical effort involved in operating with in a pertur-

bation vector

, where is the number of grid-

points and the occupied-unoccupied energy gap .

D (nm) ( )

19

We now demonstrate the performance of the stochastic meth-

od by applying it to calculate the RPA correlation energies of

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spherical cadmium-selenide (CdSe) and hydrogen passivated

silicon NCs, where the Hamiltonian is constructed from a

semiempirical pseudopotential model.32, 33 The occupied

states of the NCs were obtained using the filter diagonaliza-

tion technique34 with the implementation described in Refs. 32, 35. We used

- for approximating the step function

( ) and (see discussion of Figure 4), and

, slightly larger than half the maximal eigenvalue

range for both NCs. Various features of the NCs are summa-

rized in Table 1 and 2.

Table 2: Same as Table 1, but for hydrogen passivated silicon NCs.

D (nm) ( )

As a test, we compared the stochastic estimate and a full

summation calculation of the RPA energy for on a

point grid (see Figure 1). The deviation of the sto-

chastic approach from the exact value is within the statistical

variance for nearly all stochastic iterations. We found that for

stochastic iterations the stochastic estimate deviates

by ~ meV from the full summation value. As explained

below, the number of required stochastic iterations decreases

considerably (for a fixed error per electron) as the size of the

system increases.

Figure 2: RPA (blue) and MP2 (red) correlation energies per elec-

tron vs. the number of electrons , for silicon (top) and CdSe (bot-

tom) NCs. Insets show the statistical errors in the RPA energies,

normalized to 1000 stochastic iterations.

Figure 2 shows the RPA correlation energies for CdSe and

silicon NCs up to electrons along with a comparison

to MP2 energies obtained using the Neuhauser-Rabani-Baer

(NRB) method.26 The RPA correlation energy depends weak-

ly on the NC size in contrast to that of MP2. This is because

the NC gaps decrease with system size and MP2 energies are

sensitive to small gaps (diverging for metals). The RPA ener-

gy of silicon is somewhat above that of CdSe, and is within

the LDA bulk limit21 range of -

.

The insets of Figure 2 show the corresponding statistical er-

rors normalized to 1000 stochastic iterations. The errors de-

crease when the number of electrons in the system increases.

This shows that the algorithm profits from statistical self-

averaging. The statistical error of the CdSe NCs is approxi-

mately twice smaller than that for silicon despite having simi-

lar gaps for the same NC size. This suggests that the statisti-

cal errors are not trivially correlated with the gap.

Figure 3 shows the total CPU time for calculations that yield

a statistical error of meV per electron. The method

scales, at most, quadratically with system size but in practice,

due to self-averaging, requires considerably less statistical

sampling as the system grows and the resulting performance

is close to linear scaling. Furthermore, in comparison, the

RPA CPU time for the same statistical error is an order of

magnitude smaller than the CPU time required for the MP2

calculations. Regarding memory requirements, for the RPA

scales quadratically with system size (17GB for the largest

silicon NC) and linearly for MP2.

Figure 3: The CPU times for achieving a statistical error of meV per electron for the RPA and MP2 calculations of CdSe

NCs.

In some cases, the Chebyshev interpolation suffers from in-

stabilities. This is shown in the Figure 4 for the NC

where we plot the RPA energy estimate as function of the

length of the Chebyshev expansion. The results (dashed line)

clearly diverge as the Chebyshev expansion length grows. To

alleviate this problem we project out from ( )

and

( )

, after each operation of , the occupied orbitals ,

- where is a small system-dependent

integer.

Figure 4 shows the RPA correlation energy as a function of

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the Chebyshev interpolation order ( ). For the RPA

energy with ( ) and without projection are nearly iden-

tical (to within statistical error). For higher interpolation or-

ders than shown the RPA correlation energy without projec-

tion diverges while the projected one remains stable. For

the Chebyshev polynomial visibly diverges already

when the order is larger than 300 but becomes stable when

projection is used with . Here too, the difference be-

tween the projected and non-projected correlation energies

(before divergence) is smaller than the statistical fluctuations,

which is large in the diverged result due to the small number

of stochastic iterations done.

Figure 4: The RPA correlation energy as function of the length of

the Chebyshev interpolation polynomial for silicon NCs. Dashed

lines are the results without projection while the solid lines are for

for and for .

The present method is not only useful to calculate the RPA

energy per electron. In fact, it can also be used to obtain, for

example, the RPA corrections to chemical reaction energy

profiles or to adsorption energies. In this respect, we have

developed an approach, based on a combination of adiabatic

evolution and correlated sampling to calculate energy differ-

ences between two structural configurations.36 Preliminary

results for (enabling comparison to the full summation

result) and , moving one atom by 0.01 , indicate that

the statistical error of the energy gradient (i.e., the force due

the RPA energy) is of the comparable to the statistical error

of the total RPA energy per electron (much smaller than the

statistical error in the total energy). For and the

statistical error per stochastic iteration37 in the RPA energy is

and , respectively, compared to the statistical

error in the RPA energy per electron and ,

respectively.

Summarizing, we have presented a new stochastic approach

for calculating RPA energies for large electronic systems of

exceptional size. The method scales formally as ( ) in

terms of memory and CPU time but due to self-averaging has

a near-linear scaling CPU time performance. We calculated

the RPA correlation energy for CdSe and silicon NCs up to

diameters of with over electrons. The stochastic

approach developed here bloats, by orders of magnitude, the

size of systems that can be treated using RPA theory.

DN was supported by the DOE MEEM center, award DE-

SC0001342. RB was supported by the US-Israel Binational

Foundation (BSF). RB and ER gratefully thank the Israel

Science Foundation, grant numbers 1020/10 and 611/11, re-

spectively.

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24 (2012). 27 The numerical effort of the stochastic RPA calculation scales

quadratically with system size provided the numerical effort

involved in applying the underlying Hamiltonian to a wave

function scales linearly with system size. This happens when the

underlying Hamiltonian is derived from KS DFT since the

potential is local. Furthermore, it will also be true for a

generalized KS DFT Hamiltonian having short range exchange.

When long-range exchange is present in the Hamiltonian, it is

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function in a linear scaling way. 28 We denote occupied states by indices i,j,k,l=1,..,N and

unoccupied states by a,b,c,d=N+1,…,M; the indices

p,q,s,t=1,…,M denote "unrestricted" states. 29 D. Neuhauser and R. Baer, J. Chem. Phys. 123, 204105 (2005). 30 S. Baroni, S. de Gironcoli, A. Dal Corso, and P. Giannozzi, Rev.

Mod. Phys. 73, 515 (2001). 31 The occupied-occupied eigenvalues, although affecting R will

not affect the RPA correlation energy calculated by Eq. (3). A

proof will be given in a future publication. 32 E. Rabani, B. Hetenyi, B. J. Berne, and L. E. Brus, J. Chem.

Phys. 110, 5355 (1999). 33 L. W. Wang and A. Zunger, J. Phys. Chem. 98, 2158 (1994). 34 M. R. Wall and D. Neuhauser, J. Chem. Phys. 102, 8011 (1995). 35 S. Toledo and E. Rabani, J. Comp. Phys. 180, 256 (2002). 36 Deatails of this approach will be described in a future

publication. 37 To estimate the statistical error for I stocahstic iterations divide

by the square root of I.

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